K11a86

K11a85

K11a87

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	K11a86 Quick Notes

K11a86 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,10,19,9 X2,11,3,12 X20,13,21,14 X6,16,7,15 X8,18,9,17 X22,19,1,20 X14,21,15,22
Gauss code	1, -6, 2, -1, 3, -8, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -10
Dowker-Thistlethwaite code	4 10 12 16 18 2 20 6 8 22 14
Conway Notation	[4,211,21]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a86/ThurstonBennequinNumber Hyperbolic Volume 13.5574 A-Polynomial See Data:K11a86/A-polynomial

[edit Notes for K11a86's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -2

[edit Notes for K11a86's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-12t^{2}+18t-19+18t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 91, 2 }
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-10q^{4}+13q^{3}-14q^{2}+14q-12+9q^{-1}-5q^{-2}+3q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-14z^{4}a^{-2}+4z^{4}a^{-4}+9z^{4}-3a^{2}z^{2}-15z^{2}a^{-2}+5z^{2}a^{-4}+12z^{2}-a^{2}-5a^{-2}+2a^{-4}+5$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}+3az^{9}+7z^{9}a^{-1}+4z^{9}a^{-3}+3a^{2}z^{8}+9z^{8}a^{-2}+6z^{8}a^{-4}+6z^{8}+a^{3}z^{7}-9az^{7}-20z^{7}a^{-1}-4z^{7}a^{-3}+6z^{7}a^{-5}-13a^{2}z^{6}-36z^{6}a^{-2}-9z^{6}a^{-4}+5z^{6}a^{-6}-35z^{6}-4a^{3}z^{5}+3az^{5}+14z^{5}a^{-1}-2z^{5}a^{-3}-6z^{5}a^{-5}+3z^{5}a^{-7}+17a^{2}z^{4}+45z^{4}a^{-2}+7z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}+49z^{4}+4a^{3}z^{3}+5az^{3}-2z^{3}a^{-1}+z^{3}a^{-3}+z^{3}a^{-5}-3z^{3}a^{-7}-8a^{2}z^{2}-26z^{2}a^{-2}-5z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}-26z^{2}-a^{3}z-2az-za^{-1}+za^{-5}+za^{-7}+a^{2}+5a^{-2}+2a^{-4}+5$
The A2 invariant	$-q^{12}+q^{8}+3q^{4}-q^{2}+1+q^{-2}-2q^{-4}+3q^{-6}-3q^{-8}+q^{-10}-q^{-12}-q^{-14}+2q^{-16}-q^{-18}+q^{-20}$
The G2 invariant	$q^{60}-2q^{58}+5q^{56}-9q^{54}+10q^{52}-11q^{50}+3q^{48}+14q^{46}-35q^{44}+57q^{42}-66q^{40}+48q^{38}-8q^{36}-54q^{34}+115q^{32}-153q^{30}+144q^{28}-77q^{26}-25q^{24}+131q^{22}-196q^{20}+197q^{18}-123q^{16}+13q^{14}+97q^{12}-163q^{10}+159q^{8}-78q^{6}-23q^{4}+116q^{2}-144+96q^{-2}-2q^{-4}-111q^{-6}+188q^{-8}-199q^{-10}+135q^{-12}-10q^{-14}-126q^{-16}+233q^{-18}-263q^{-20}+201q^{-22}-81q^{-24}-64q^{-26}+176q^{-28}-222q^{-30}+186q^{-32}-84q^{-34}-29q^{-36}+115q^{-38}-137q^{-40}+82q^{-42}-82q^{-46}+119q^{-48}-98q^{-50}+32q^{-52}+51q^{-54}-118q^{-56}+147q^{-58}-124q^{-60}+63q^{-62}+7q^{-64}-72q^{-66}+110q^{-68}-116q^{-70}+101q^{-72}-58q^{-74}+14q^{-76}+30q^{-78}-62q^{-80}+71q^{-82}-66q^{-84}+48q^{-86}-20q^{-88}-4q^{-90}+22q^{-92}-30q^{-94}+28q^{-96}-20q^{-98}+11q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

Further Quantum Invariants

K11a86 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a86"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{4}+5t^{3}-12t^{2}+18t-19+18t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}-2z^{4}-z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 91, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{7}-3q^{6}+6q^{5}-10q^{4}+13q^{3}-14q^{2}+14q-12+9q^{-1}-5q^{-2}+3q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-14z^{4}a^{-2}+4z^{4}a^{-4}+9z^{4}-3a^{2}z^{2}-15z^{2}a^{-2}+5z^{2}a^{-4}+12z^{2}-a^{2}-5a^{-2}+2a^{-4}+5$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{10}a^{-2}+z^{10}+3az^{9}+7z^{9}a^{-1}+4z^{9}a^{-3}+3a^{2}z^{8}+9z^{8}a^{-2}+6z^{8}a^{-4}+6z^{8}+a^{3}z^{7}-9az^{7}-20z^{7}a^{-1}-4z^{7}a^{-3}+6z^{7}a^{-5}-13a^{2}z^{6}-36z^{6}a^{-2}-9z^{6}a^{-4}+5z^{6}a^{-6}-35z^{6}-4a^{3}z^{5}+3az^{5}+14z^{5}a^{-1}-2z^{5}a^{-3}-6z^{5}a^{-5}+3z^{5}a^{-7}+17a^{2}z^{4}+45z^{4}a^{-2}+7z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}+49z^{4}+4a^{3}z^{3}+5az^{3}-2z^{3}a^{-1}+z^{3}a^{-3}+z^{3}a^{-5}-3z^{3}a^{-7}-8a^{2}z^{2}-26z^{2}a^{-2}-5z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}-26z^{2}-a^{3}z-2az-za^{-1}+za^{-5}+za^{-7}+a^{2}+5a^{-2}+2a^{-4}+5$

Vassiliev invariants

V2 and V3:

(-1, -2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-4$ $-16$ $8$ ${\frac {34}{3}}$ ${\frac {62}{3}}$ $64$ ${\frac {608}{3}}$ ${\frac {224}{3}}$ $80$ $-{\frac {32}{3}}$ $128$ $-{\frac {136}{3}}$ $-{\frac {248}{3}}$ ${\frac {13169}{30}}$ ${\frac {714}{5}}$ ${\frac {898}{45}}$ ${\frac {1999}{18}}$ $-{\frac {2191}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$2 is the signature of K11a86. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     2   -2 11                   4 1   3 9                 6 2     -4 7               7 4       3 5             7 6         -1 3           7 7           0 1         6 8             2 -1       3 6               -3 -3     2 6                 4 -5   1 3                   -2 -7   2                     2 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[11, Alternating, 86]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 86]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 8, 17, 7], X[18, 10, 19, 9], X[2, 11, 3, 12], X[20, 13, 21, 14], X[6, 16, 7, 15], X[8, 18, 9, 17], X[22, 19, 1, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 86]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -10]
In[5]:=	BR[Knot[11, Alternating, 86]]
Out[5]=	BR[Knot[11, Alternating, 86]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 86]][t]
Out[6]=	-4 5 12 18 2 3 4 -19 - t + -- - -- + -- + 18 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 86]][z]
Out[7]=	2 4 6 8 1 - z - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 86]}
In[9]:=	{KnotDet[Knot[11, Alternating, 86]], KnotSignature[Knot[11, Alternating, 86]]}
Out[9]=	{91, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 86]][q]
Out[10]=	-4 3 5 9 2 3 4 5 6 -12 - q + -- - -- + - + 14 q - 14 q + 13 q - 10 q + 6 q - 3 q + 3 2 q q q 7 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 86], Knot[11, Alternating, 205]}
In[12]:=	A2Invariant[Knot[11, Alternating, 86]][q]
Out[12]=	-12 -8 3 -2 2 4 6 8 10 12 14 1 - q + q + -- - q + q - 2 q + 3 q - 3 q + q - q - q + 4 q 16 18 20 2 q - q + q
In[13]:=	Kauffman[Knot[11, Alternating, 86]][a, z]
Out[13]=	2 2 2 5 2 z z z 3 2 z 2 z 5 + -- + -- + a + -- + -- - - - 2 a z - a z - 26 z - -- + ---- - 4 2 7 5 a 8 6 a a a a a a 2 2 3 3 3 3 5 z 26 z 2 2 3 z z z 2 z 3 3 3 ---- - ----- - 8 a z - ---- + -- + -- - ---- + 5 a z + 4 a z + 4 2 7 5 3 a a a a a a 4 4 4 4 5 5 5 4 z 5 z 7 z 45 z 2 4 3 z 6 z 2 z 49 z + -- - ---- + ---- + ----- + 17 a z + ---- - ---- - ---- + 8 6 4 2 7 5 3 a a a a a a a 5 6 6 6 14 z 5 3 5 6 5 z 9 z 36 z 2 6 ----- + 3 a z - 4 a z - 35 z + ---- - ---- - ----- - 13 a z + a 6 4 2 a a a 7 7 7 8 8 6 z 4 z 20 z 7 3 7 8 6 z 9 z 2 8 ---- - ---- - ----- - 9 a z + a z + 6 z + ---- + ---- + 3 a z + 5 3 a 4 2 a a a a 9 9 10 4 z 7 z 9 10 z ---- + ---- + 3 a z + z + --- 3 a 2 a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 86]], Vassiliev[3][Knot[11, Alternating, 86]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[11, Alternating, 86]][q, t]
Out[15]=	3 1 2 1 3 2 6 3 8 q + 7 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t 6 6 q 3 5 5 2 7 2 7 3 9 3 --- + --- + 7 q t + 7 q t + 6 q t + 7 q t + 4 q t + 6 q t + q t t 9 4 11 4 11 5 13 5 15 6 2 q t + 4 q t + q t + 2 q t + q t